numerical solution of the wave equation

TPG�� Anvendt datateknikk i geofag og petroleumsteknologi�ving �� Matlab

�

Numerical solution of the wave equation

Introduction

Marine seismic oil�gas exploration is carried out by generating soundwaves thatpropagate through the earth� The waves are reected from interfaces between dierentgeological layers� Receivers are recording the reected waves� converting the acousticenergy into electronic energy� and then storing the data on tape� By dierent seismicprocessing techniques an image of the subsurface can be made from the acquired data�

In this exercise we will be simulating propagation of sound in simple media using the�nite dierence method� The �nite dierence method is to be implemented in Matlab�and also the plots should be made in Matlab�

To avoid running out of computer memory we will restrict ourselves to two spatialdimensions represented by the x�axis and the z�axis� The equation de�ning the wavepropagation of the pressure �eld P x� z� t� is a partial dierential equation given by

��P x� z� t�

�x��

��P x� z� t�

�z��

�

c� x� z�

��P x� z� t�

�t�� S x� z� t� ��

where c x� z� is the wave�velocity and S x� z� t� is a source termWe shall replace the dierential equation �� with a �nite dierence equation being anapproximation to equation �� The �nite dierence equation can be rearranged� enablingus to �nd the pressure at the next time step as a function of the pressure at present andprevious timesteps�

Derivation of the �nite di�erence formula

As a starting point for deriving the �nite dierence equation we consider the de�nition ofthe derivative of P x� z� t� with respect to x�

�P x� z� t�

�x� lim

�x��

P x��x� z� t�� P x� z� t�

�x

Allowing �x to be �nite rather than in�nitesimal yields the approximation�

�P x� z� t�

�x�

P x��x� z� t�� P x� z� t�

�x

The second derivative is obtained by repeating the above procedure�

Norges teknisk�naturvitenskapelige universitetInstitutt for petroleumsteknologi og anvendt geofysikk

Stipendiat Tommy Toverud


�

��P x� z� t�

�x��

�

�x

�P x� z� t�

�x�

�P �x��x�z�t��x

��P �x�z�t�

�x

�x�

P x� ��x� z� t�� P x��x� z� t�� P x��x� z� t�� P x� z� t��

�x�

we may perform a backward shift �x resulting in�

��P x� z� t�

�x��

P x��x� z� t�� P x� z� t� � P x��x� z� t�

�x�

We introduce a more compact notation�

P x� z� t� � P nl�m � P l�x�m�z� n�t� ��

c x� z� � cl�m � c l�x�m�z� ��

S x� z� t� � Snl�m � S l�x�m�z� n�t� ��

��

On �nite dierence form equation �� appears this way�

P nl��m � �P n

l�m � P nl��m

�x��

P nl�m�� P n

l�m � P nl�m��

�z��

�

c�l�m

P n��l�m � �P n

l�m � P n��l�m

�t�� Sn

l�m

��

For simplicity we use the same step�length in the x� and z�direction �z � �x� Henceequation �� may be rewritten�

P n��l�m �

c�l�m�t�

�x�

�P nl��m � P n

l�m�� P nl�m � P n

l��m � P nl�m��

�� P n

l�m � P n��l�m � c�l�m�t�Sn

l�m

��

Part ��

a� Make a Matlab program that gives the pressure �eld as a function of x and z at time�� s� n�� and plot the result� Use equation ��Let �x � �y � �m and let �t � ��s� m and l should cover a ��x�� grid�area�Assume a constant wave velocity of c� � �� m�s The velocity of sound in water��

It is important that c�m�t�

�x��

�Otherwise the error will increase for each timestep

causing instability��Boundary conditions are P n

��m � P n��m � P n

l�� P nl�� while the initial conditions are

given by

P �l�m � � ��

P �l�m � � ��




�

n−1

n

n+1

Figure �� Finite dierence grid� The pressure at the star is calculated from the values ofthe pressure at the dots

To generate the waves we use a source Snl�m de�ned by

Sn�� f n� ��

Snl�m � � else ��

f n� may be down�loaded from http��www�ipt�ntnu�no��bufag�kilde� Save kildeon your computer and load into Matlab with the command load kilde� Matlabfunctions suitable for two�dimensional plots include mesh� contour and contourf� Ifyou have named the pressure P� you could use the command contour P��You may try to �nd the pressure at other timesteps� but storing all timesteps in memorywill at some point cause memory shortage�

b� Register the pressure at all timesteps and all x�coordinates� but �x the z�coordinate atm�� Make a plot of this as well� This simulates seismic acquisition along the linem��

Part ��

Repeat part �a� and b�� but insert a reector along m�� by letting the velocity be�� m�s above m��

Matlab commands

The pc�version of Matlab is started by clicking the matlab�icon or writing matlab in therun window� The Unix version is started from the dtterm window or xterm window with




�

the command matlab� Writing quit at the Matlab prompt will terminate the session�Documentation for all matlab commands is obtained writing help command The currentworking directory can be found with the command pwd� You may change directory to e�g�c��yourdirectory assuming pc�matlab� using the command

cd c��yourdirectory

load kilde into Matlab by the commandload c��yourdirectory�kilde

You can plot kilde using the commandplot�kilde�

plot�kilde�� plots the �rst � timesamplesplot�kilde�� plots every fourth sample up to ��plot�kilde� �� reverses kilde

Reversing kilde can also be achieved by writing plot�fliplr�kilde�� The values alongthe x�axis correspond to the component�numbers of kilde if only kilde is included in theplot command� To adjust the x�axis� we �rst create a variable

x��length�kilde��

x has the same number of components as kilde which is important�� This number isfound with the length command� The component�values are�� The semicolon prevents these values from beingdisplayed on the screen We specify the x�coordinate by issuing two input vectors for theplot command� x and kilde� Two more vectors will be interpreted as a newx�coordinate�y�coordinate pair yielding two curves

plot�x�kilde�

plot�x�kilde�x�kilde�� Two curvesVectors can be created �manually� by writing the component values between brackets

� � � ��

This way you can manipulate kilde plot�� kilde�� kilde�� The axis isspeci�ed using the command axis

axis��xmin xmax ymin ymax��

A matrix can be constructed like thismatrise��

The rows of the matrix are separated by semicolons� This can be plotted using forinstance

mesh�matrise� orcontour�matrise�

Loops are constructed similarly to other programming languages� It can be done like thisfor k�� energi�k��kilde�k��kilde�k��end

Matlab oers an alternative to the above structure�k�� energi�k��kilde�k��kilde�k��

Placing a period before the multiplication sign ensures that multiplication is carried outcomponent by component� The last alternative is much faster than the for�end alternativeConditional statements are implemented this way

if �x�� y� �end




�

x is tested for equality with zero note two equal signs�� If this is true� y is assigned thevalue zero� In the next example x has to be dierent from zero and z has to be less than� for the operation y�� to be carried out

if ��x�� z�� y� �end

�� corresponds to �and�� while �� corresponds to �or��




�

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Figure �� Wave propagation as in part �� plotted using the command sequence�clf�for f�� plot P f��f��hold on�end

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Figure �� Arrival times for dierent x values� z��




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0 50 100 150 200 250 300 350 400 450 5000

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Figure �� Wave propagation with a reector at z��

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Figure �� Arrival times for dierent x values� z�� direct wave and reection�



numerical solution of the wave equation

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